Problem : To find whether x and y belong to same group or not, i.e., to find if x and y are direct/indirect friends.

Solution : Partitioning the individuals into different sets according to the groups in which they fall. This method is known as disjoint set data structure which maintains collection of disjoint sets and each set is represented by its representative which is one of its members.

Approach:

How to Resolve sets ? Initially all elements belong to different sets. After working on the given relations, we select a member as representative. There can by many ways to select a representative, a simple one is to select with the biggest index.

Check if 2 persons are in the same group ? If representatives of two individuals are same, then they’ll become friends.

Data Structures used:

Array : An array of integers, called parent[]. If we are dealing with n items, i’th element of the array represents the i’th item. More precisely, the i’th element of the array is the parent of the i’th item. These relationships create one, or more, virtual trees.

Tree : It is a disjoint set. If two elements are in the same tree, then they are in the same disjoint set. The root node (or the topmost node) of each tree is called the representative of the set. There is always a single unique representative of each set. A simple rule to identify representative is, if i is the representative of a set, then parent[i] = i. If i is not the representative of his set, then it can be found by traveling up the tree until we find the representative.

Operations :

Find : Can be implemented by recursively traversing the parent array until we hit a node who is parent of itself.

// Finds the representative of the set that
// i is an element of
public int find(int i)
{
// If i is the parent of itself
if (parent[i] == i)
{
// Then i is the representative of
// this set
return i;
}
else
{
// Else if i is not the parent of
// itself, then i is not the
// representative of his set. So we
// recursively call Find on its parent
return find(parent[i]);
}
}

Union: It takes, as input, two elements. And finds the representatives of their sets using the find operation, and finally puts either one of the trees (representing the set) under the root node of the other tree, effectively merging the trees and the sets.

// Unites the set that includes i and
// the set that includes j
public void union(int i, int j)
{
// Find the representatives
// (or the root nodes) for the set
// that includes i
int irep = this.Find(i),
// And do the same for the set
// that includes j
int jrep = this.Find(j);
// Make the parent of i’s representative
// be j’s representative effectively
// moving all of i’s set into j’s set)
this.Parent[irep] = jrep;
}

Improvements (Union by Rank and Path Compression)

The efficiency depends heavily on the height of the tree. We need to minimize the height of tree in order improve the efficiency. We can use Path Compression and Union by rank methods to do so.

Path Compression (Modifications to find()) : It speeds up the data structure by compressing the height of the trees. It can be achieved by inserting a small caching mechanism into the Find operation. Take a look at the code for more details:

// Finds the representative of the set that i
// is an element of.
int find(int i)
{
// If i is the parent of itself
if (Parent[i] == i)
{
// Then i is the representative
return i;
}
else
{
// Recursively find the representative.
int result = find(Parent[i]);
// We cache the result by moving i’s node
// directly under the representative of this
// set
Parent[i] = result;
// And then we return the result
return result;
}
}

Union by Rank: First of all, we need a new array of integers called rank[]. Size of this array is same as the parent array. If i is a representative of a set, rank[i] is the height of the tree representing the set.
Now recall that, in the Union operation, it doesn’t matter which of the two trees is moved under the other (see last two image examples above). Now what we want to do is minimize the height of the resulting tree. If we are uniting two trees (or sets), let’s call them left and right, then it all depends on the rank of left and the rank of right.

If the rank of left is less than the rank of right, then it’s best to move left under right, because that won’t change the rank of right (while moving right under left would increase the height). In the same way, if the rank of right is less than the rank of left, then we should move right under left.

If the ranks are equal, it doesn’t matter which tree goes under the other, but the rank of the result will always be one greater than the rank of the trees.

// Unites the set that includes i and the set
// that includes j
void union(int i, int j)
{
// Find the representatives (or the root nodes)
// for the set that includes i
int irep = this.find(i);
// And do the same for the set that includes j
int jrep = this.Find(j);
// Elements are in same set, no need to
// unite anything.
if (irep == jrep)
return;
// Get the rank of i’s tree
irank = Rank[irep],
// Get the rank of j’s tree
jrank = Rank[jrep];
// If i’s rank is less than j’s rank
if (irank < jrank)
{
// Then move i under j
this.parent[irep] = jrep;
}
// Else if j’s rank is less than i’s rank
else if (jrank < irank)
{
// Then move j under i
this.Parent[jrep] = irep;
}
// Else if their ranks are the same
else
{
// Then move i under j (doesn’t matter
// which one goes where)
this.Parent[irep] = jrep;
// And increment the the result tree’s
// rank by 1
Rank[jrep]++;
}
}

// A Java program to implement Disjoint Set Data
// Structure.
import java.io.*;
import java.util.*;
class DisjointUnionSets
{
int[] rank, parent;
int n;
// Constructor
public DisjointUnionSets(int n)
{
rank = new int[n];
parent = new int[n];
this.n = n;
makeSet();
}
// Creates n sets with single item in each
void makeSet()
{
for (int i=0; i<n; i++)
{
// Initially, all elements are in
// their own set.
parent[i] = i;
}
}
// Returns representative of x's set
int find(int x)
{
// Finds the representative of the set
// that x is an element of
if (parent[x]!=x)
{
// if x is not the parent of itself
// Then x is not the representative of
// his set,
parent[x] = find(parent[x]);
// so we recursively call Find on its parent
// and move i's node directly under the
// representative of this set
}
return parent[x];
}
// Unites the set that includes x and the set
// that includes x
void union(int x, int y)
{
// Find representatives of two sets
int xRoot = find(x), yRoot = find(y);
// Elements are in the same set, no need
// to unite anything.
if (xRoot == yRoot)
return;
// If x's rank is less than y's rank
if (rank[xRoot] < rank[yRoot])
// Then move x under y so that depth
// of tree remains less
parent[xRoot] = yRoot;
// Else if y's rank is less than x's rank
else if (rank[yRoot] < rank[xRoot])
// Then move y under x so that depth of
// tree remains less
parent[yRoot] = xRoot;
else // if ranks are the same
{
// Then move y under x (doesn't matter
// which one goes where)
parent[yRoot] = xRoot;
// And increment the the result tree's
// rank by 1
rank[xRoot] = rank[xRoot] + 1;
}
}
}
// Driver code
public class Main
{
public static void main(String[] args)
{
// Let there be 5 persons with ids as
// 0, 1, 2, 3 and 4
int n = 5;
DisjointUnionSets dus =
new DisjointUnionSets(n);
// 0 is a friend of 2
dus.union(0, 2);
// 4 is a friend of 2
dus.union(4, 2);
// 3 is a friend of 1
dus.union(3, 1);
// Check if 4 is a friend of 0
if (dus.find(4) == dus.find(0))
System.out.println("Yes");
else
System.out.println("No");
// Check if 1 is a friend of 0
if (dus.find(1) == dus.find(0))
System.out.println("Yes");
else
System.out.println("No");
}
}

Try to solve this problem and check how much you learnt and do comment on the complexity of the given question.

This article is contributed by Nikhil Tekwani. If you like GeeksforGeeks and would like to contribute, you can also write an article and mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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